Burmilla Health - As mentioned in the post on the history and origins of this cat breed, the Burmilla's founding cats are the Burmese and Persian (Chinchilla coat). The Persian cat breed is known to suffer from a number of disorders including the genetic disorder PKD - Polycystic Kidney Disease (apparently about one third of all purebred Persians contract PKD).
This disease is nasty. Cysts form in the kidneys. It can lead to renal failure, which kills as you might expect.
Accordingly, the Burmilla may have inherited this disease. Good breeders will screen for it by DNA testing. This ensures as near as possible that the breeding program operated by the breeder is free of this disease. Enquire about this with the breeder. They should have medical documentation certifying that their cats are free of PKD.
Genetic Defect USA Burmese
As for the Burmese cat the other founding parent cat of the Burmilla, it might be useful to mention that the USA Burmese carries a genetic defect that occasionally rears its ugly head in the form of the birth of kittens with deformed heads - quite tragic really. These kittens are killed. I do not know how this gene is or isn't transmitted to the Burmilla. It is probably transmitted but is recessive so is rarely seen. I'd ask about and watch for this as well, though. This disorder does not affect UK Burmese cats apparently.
Burmilla Health to Home Page